A004130 -id:A004130 - cp's OEIS frontend

A093544 Numerator of (4*n-3)/A000265(n). Numerator of pairwise quotients of A004130.

1, 5, 3, 13, 17, 7, 25, 29, 11, 37, 41, 15, 49, 53, 19, 61, 65, 23, 73, 77, 27, 85, 89, 31, 97, 101, 35, 109, 113, 39, 121, 125, 43, 133, 137, 47, 145, 149, 51, 157, 161, 55, 169, 173, 59, 181, 185, 63, 193, 197, 67, 205, 209, 71, 217, 221, 75, 229, 233, 79, 241

Offset: 1

Ralf Stephan, Mar 31 2004

Terms are a rearrangement of the odd numbers not of form 12*k+9.

Also given by the following linear quasi-polynomial mod 3: (4 n - 3) If[Mod[n, 3] == 0, 1/3, 1]. - Eric Rowland, Feb 24 2009

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Table[(4 n - 3) If[Mod[n, 3] == 0, 1/3, 1], {n, 61}] (* Eric Rowland, Feb 24 2009 *)

a(n)=numerator((4*n-3)/n*2^valuation(n,2))

A123854 Denominators in an asymptotic expansion for the cubic recurrence sequence A123851.

Original entry on oeis.org

1, 4, 32, 128, 2048, 8192, 65536, 262144, 8388608, 33554432, 268435456, 1073741824, 17179869184, 68719476736, 549755813888, 2199023255552, 140737488355328, 562949953421312, 4503599627370496, 18014398509481984, 288230376151711744, 1152921504606846976

Offset: 0

Petros Hadjicostas and Jonathan Sondow, Oct 15 2006

A cubic analog of the asymptotic expansion A116603 of Somos's quadratic recurrence sequence A052129. Numerators are A123853.
Equals 2^A004134(n); also the denominators in expansion of (1-x)^(-1/4). - Alexander Adamchuk, Oct 27 2006
All terms are powers of 2 and log_2 a(n) = A004134(n) = 3*n - A000120(n). - Alexander Adamchuk, Oct 27 2006 [Edited by Petros Hadjicostas, May 14 2020]
Is this the same sequence as A088802? - N. J. A. Sloane, Mar 21 2007
Almost certainly this is the same as A088802. - Michael Somos, Aug 23 2007
Denominators of Gegenbauer_C(2n,1/4,2). The denominators of Gegenbauer_C(n,1/4,2) give the doubled sequence. - Paul Barry, Apr 21 2009
If the Greubel formula in A088802 and the Luschny formula here are correct (they are the same), the sequence is a duplicate of A088802. - R. J. Mathar, Aug 02 2023

			A123851(n) ~ c^(3^n)*n^(- 1/2)/(1 + 3/(4*n) - 15/(32*n^2) + 113/(128*n^3) - 5397/(2048*n^4) + ...) where c = 1.1563626843322... is the cubic recurrence constant A123852.

Steven R. Finch, Mathematical Constants, Cambridge University Press, Cambridge, 2003, p. 446.

G. C. Greubel, Table of n, a(n) for n = 0..1000
T. M. Apostol, On the Lerch zeta function, Pacific J. Math. 1 (1951), 161-167. [In Eq. (3.7), p. 166, the index in the summation for the Apostol-Bernoulli numbers should start at s = 0, not at s = 1. - _Petros Hadjicostas_, Aug 09 2019]
Jonathan Sondow and Petros Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, arXiv:math/0610499 [math.CA], 2006.
Jonathan Sondow and Petros Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, J. Math. Anal. Appl. 332 (2007), 292-314.
Eric Weisstein's World of Mathematics, Somos's Quadratic Recurrence Constant.
Aimin Xu, Asymptotic expansion related to the Generalized Somos Recurrence constant, International Journal of Number Theory 15(10) (2019), 2043-2055. [The author gives recurrences and other formulas for the coefficients of the asymptotic expansion using the Apostol-Bernoulli numbers (see the reference above) and the Bell polynomials. - _Petros Hadjicostas_, Aug 09 2019]

Cf. A052129, A112302, A116603, A123851, A123852, A123853 (numerators).

Cf. A004134, A004130, A000120.

f:=proc(t,x) exp(sum(ln(1+m*x)/t^m,m=1..infinity)); end; for j from 0 to 29 do denom(coeff(series(f(3,x),x=0,30),x,j)); od;
# Alternatively:
A123854 := n -> denom(binomial(1/4,n)):
seq(A123854(n), n=0..25); # Peter Luschny, Apr 07 2016

Denominator[CoefficientList[Series[ 1/Sqrt[Sqrt[1-x]], {x, 0, 25}], x]] (* Robert G. Wilson v, Mar 23 2014 *)

vector(25, n, n--; denominator(binomial(1/4,n)) ) \\ G. C. Greubel, Aug 08 2019

# uses[A000120]
def A123854(n): return 1 << (3*n-A000120(n))
[A123854(n) for n in (0..25)]  # Peter Luschny, Dec 02 2012

From Alexander Adamchuk, Oct 27 2006: (Start)
a(n) = 2^A004134(n).
a(n) = 2^(3n - A000120(n)). (End)
a(n) = denominator(binomial(1/4,n)). - Peter Luschny, Apr 07 2016

A004134 Denominators in expansion of (1-x)^{-1/4} are 2^a(n).

Original entry on oeis.org

0, 2, 5, 7, 11, 13, 16, 18, 23, 25, 28, 30, 34, 36, 39, 41, 47, 49, 52, 54, 58, 60, 63, 65, 70, 72, 75, 77, 81, 83, 86, 88, 95, 97, 100, 102, 106, 108, 111, 113, 118, 120, 123, 125, 129, 131, 134, 136, 142, 144, 147, 149, 153, 155, 158, 160, 165, 167, 170, 172, 176, 178

Offset: 0

N. J. A. Sloane

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
R. Stephan, Some divide-and-conquer sequences ...
R. Stephan, Table of generating functions

Cf. A004130.

Cf. A005187.

Log2[ Denominator[ CoefficientList[ Series[ 1/Sqrt[Sqrt[1 - x]], {x, 0, 61}], x]]] (* Robert G. Wilson v, Mar 23 2014 *)
f[n_] := 3 n - DigitCount[n, 2, 1]; Array[f, 62, 0] (* or *)
a[n_] := If[ OddQ@ n, a[(n - 1)/2] + 3 (n - 1)/2 + 2, a[n/2] + 3 n/2]; a[0] = 0; Array[a, 62, 0] (* Robert G. Wilson v, Mar 23 2014 *)

{a(n) = if( n<0, 0, 3*n - subst( Pol( binary( n ) ), x, 1) ) } /* Michael Somos, Aug 23 2007 */

a(n) = 3*n - hammingweight(n); \\ Joerg Arndt, Mar 23 2014

a(n) = 3*n - A000120(n). Recurrence: a(2n) = a(n) + 3n, a(2n+1) = a(n) + 3n + 2. Proved by Mitch Harris, following a conjecture by Ralf Stephan.

a(n) = A005187(n) + n. - Cyril Damamme, Aug 04 2015

A364660 Numerators of coefficients in expansion of (1 + x)^(1/4).

Original entry on oeis.org

1, 1, -3, 7, -77, 231, -1463, 4807, -129789, 447051, -3129357, 11094993, -159028233, 574948227, -4188908511, 15359331207, -906200541213, 3358272593907, -25000473754641, 93422822977869, -1401342344668035, 5271716439465465, -39777496770512145, 150462705175415505, -4564035390320936985

Offset: 0

Ilya Gutkovskiy, Aug 01 2023

			(1 + x)^(1/4) = 1 + x/4 - 3*x^2/32 + 7*x^3/128 - 77*x^4/2048 + 231*x^5/8192 - 1463*x^6/65536 + ...
Coefficients are 1, 1/4, -3/32, 7/128, -77/2048, 231/8192, -1463/65536, ...

Denominators are A088802, A123854.

Cf. A002596, A004130, A008545, A067622, A364661.

nmax = 24; CoefficientList[Series[(1 + x)^(1/4), {x, 0, nmax}], x] // Numerator
Table[Binomial[1/4, n], {n, 0, 24}] // Numerator

my(x='x+O('x^30)); apply(numerator, Vec((1 + x)^(1/4))) \\ Michel Marcus, Aug 02 2023

A181161 Numerator in abs(binomial(-1/8,n)).

Original entry on oeis.org

1, 1, 9, 51, 1275, 8415, 115005, 805035, 45886995, 331406075, 4838528695, 35629165845, 1056998586735, 7886835608715, 118302534130725, 891212423784795, 107836703277960195, 818290277815109715

Offset: 0

Emanuele Munarini, Jan 25 2011

nonn

Michael De Vlieger, Table of n, a(n) for n = 0..834
Paveł Szabłowski, Beta distributions whose moment sequences are related to integer sequences listed in the OEIS, Contrib. Disc. Math. (2024) Vol. 19, No. 4, 85-109. See p. 93.

A004130

w[n_] := Numerator[Binomial[-1/8, n] (-1)^n];
Table[w[n], {n, 0, 12}]

a(n) = 2^(e_2((2*n)!)-n)/n! * Product[8k+1,{k,0,n-1}] where e_2((2n)!) is the highest power of 2 that divides (2*n)! (sequence A005187)

cp's OEIS Frontend

A093544 Numerator of (4*n-3)/A000265(n). Numerator of pairwise quotients of A004130.

Views

Author

Keywords

Comments

Links

Programs

Mathematica

PARI

A123854 Denominators in an asymptotic expansion for the cubic recurrence sequence A123851.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Sage

Formula

A004134 Denominators in expansion of (1-x)^{-1/4} are 2^a(n).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A364660 Numerators of coefficients in expansion of (1 + x)^(1/4).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

A181161 Numerator in abs(binomial(-1/8,n)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula